interval

Loosely speaking, an interval is a part of the real numbers
that start at one number and stops at another number. For instance,
all numbers greater that 1 and smaller than 2 form in interval.
Another interval is formed by numbers greater or equal to 1 and
smaller than 2. Thus, when talking about intervals, it is necessary
to specify whether the endpoints are part of the interval or not.
There are then four types of intervals with three different names:
open, closed and half-open.
Let us next define these precisely.

1.

The open interval contains neither of the endpoints.
If a and b are real numbers, then the open interval of
numbers between a and b is written as (a,b) and

(a,b)={x∈ℝ∣a<x<b}.

2.

The closed interval contains both endpoints.
If a and b are real numbers, then the closed interval
is written as [a,b] and

[a,b]={x∈ℝ∣a≤x≤b}.

3.

A half-open interval contains only one of
the endpoints.
If a and b are real numbers, the half-open intervals(a,b] and [a,b) are defined as

(a,b]

=

{x∈ℝ∣a<x≤b},

⁢[a,b)

=

{x∈ℝ∣a≤x<b}.

Note that this definition includes the empty set as an interval by, for example, taking the interval (a,a) for any a.

An interval is a subset S of a totally ordered setT with the property that whenever x and y are in S and x<z<y then z is in S. Applied to the real numbers, this encompasses open, closed, half-open, half-infinite, infinite, empty, and one-point intervals. All the various different types of interval in ℝ have this in common. Intervals in ℝ are connected under the usual topology.

There is a standard way of graphically representing intervals
on the real line using filled and empty circles. This is illustrated in
the below figures:

The logic is here that a empty circle represent a point not belonging
to the interval, while a filled circle represents a point belonging
to the interval. For example, the first interval is an open interval.

Infinite intervals

If we allow either (or both) of a and b to be infinite, then we
define

(a,∞)

=

{x∈ℝ∣x>a},

⁢[a,∞)

=

{x∈ℝ∣x≥a},

(-∞,a)

=

{x∈ℝ∣x<a},

(-∞,a]

=

{x∈ℝ∣x≤a},

(-∞,∞)

=

ℝ.

The graphical representation of infinite intervals is as follows:

Note on naming and notation

In [1, 2], an open interval is always called
a segment, and a closed interval is called simply an interval.
However, the above naming with open, closed, and half-open interval seems
to be more widely adopted.
See e.g. [3, 4, 5]. To distinguish between [a,b) and
(a,b], the former is sometimes called a right half-open interval and
the latter a left half-open interval[6].
The notation (a,b), [a,b), (a,b], [a,b] seems to be standard. However,
some authors (especially from the French school) use notation
]a,b[, [a,b[, ]a,b], [a,b] instead of the above (in the same ). Bourbaki, for example, uses this notation.

This entry contains content adapted from the Wikipedia article http://en.wikipedia.org/wiki/Interval_(mathematics)Interval (mathematics) as of November 10, 2006.